On the Nowicki Conjecture for the free Lie algebra of rank 2

We consider the free Lie algebra generated by two elements x and y, namely L(x, y) under the action of the Weitzenbock derivation {\delta} sending y to x and x to 0 and we present its algebra of constants as the algebra generated by the integrals of the relations among the generators of a proper subalgebra of L(x, y).